Free trees and the optimal bound in Wehrung's theorem

We prove that there is a distributive (V, 0, 1)-semilattice G of size N-2 such that there is no weakly distributive (V, 0)-homomorphism from Con(c) A to G with 1 in its range, for any algebra A with either a congruence-compatible structure of a (V, 1)-semilattice or a congruence-compatible structure of a lattice. In particular, G is not isomorphic to the (V, 0)-semilattice of compact congruences of any lattice. This improves Wehrung's solution of Dilworth's Congruence Lattice Problem, by giving the best cardinality bound possible. The main ingredient of our proof is the modification of Kuratowski's Free Set Theorem, which involves what we call free trees.